Integer Multiplication is Commutative
Theorem
The operation of multiplication on the set of integers $\Z$ is commutative:
- $\forall x, y \in \Z: x \times y = y \times x$
Proof
From the formal definition of integers, $\eqclass {a, b} {}$ is an equivalence class of ordered pairs of natural numbers.
Let $x = \eqclass {a, b} {}$ and $y = \eqclass {c, d} {}$ for some $x, y \in \Z$.
Then:
| \(\ds x \times y\) | \(=\) | \(\ds \eqclass {a, b} {}\times \eqclass {c, d} {}\) | Definition of Integer | |||||||||||
| \(\ds \) | \(=\) | \(\ds \eqclass {a c + b d, a d + b c} {}\) | Definition of Integer Multiplication | |||||||||||
| \(\ds \) | \(=\) | \(\ds \eqclass {c a + d b, d a + c b} {}\) | Natural Number Multiplication is Commutative | |||||||||||
| \(\ds \) | \(=\) | \(\ds \eqclass {c a + d b, c b + d a} {}\) | Natural Number Addition is Commutative | |||||||||||
| \(\ds \) | \(=\) | \(\ds \eqclass {c, d} {} \times \eqclass {a, b} {}\) | Definition of Integer Multiplication | |||||||||||
| \(\ds \) | \(=\) | \(\ds y \times x\) | Definition of Integer |
$\blacksquare$
Sources
- 1951: Nathan Jacobson: Lectures in Abstract Algebra: Volume $\text { I }$: Basic Concepts ... (previous) ... (next): Introduction $\S 5$: The system of integers
- 1964: W.E. Deskins: Abstract Algebra ... (previous) ... (next): $\S 2.5$: Theorem $2.23: \ \text{(i)}$
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $20$. The Integers: Theorem $20.10$
- 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): Chapter $2$: Integers and natural numbers: $\S 2.1$: The integers: $\mathbf Z. \, 2$
- 1994: H.E. Rose: A Course in Number Theory (2nd ed.) ... (previous) ... (next): $1$ Divisibility: $1.1$ The Euclidean algorithm and unique factorization
- 2008: Paul Halmos and Steven Givant: Introduction to Boolean Algebras ... (previous) ... (next): $\S 1$